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\title{Equilibria in Simultaneous Auctions: Algorithms and Theory}
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\section{Contributions}
In this paper, we:
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\item analyse equilibrium strategies for agents participating in multiple, simultaneously-held sealed-bid auctions with discrete bids
\item do so theoretically for a limited setting (2 bidders, 2 auctions, 2 bid levels per auction), but for various preference relations (substitutes, complements, and inbetween)
\item show how this analysis can be extended to other cases, but becomes intractable very quickly
\item introduce computational algorithm based on fictitious play (FP)
\item analyse the simple setting using FP
\item show that theoretical results correspond to FP results
\item analyse more general settings using FP
\item show that FP converges, i.e. finds an equilibrium
\item compare different auction formats (first and second price)
\item discuss the application of the algorithm as well as theoretical approach to more general setting, i.e. any Bayesian game where players have quasi-linear utility functions and finite possible actions 
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\section{Motivation}
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\item our analysis applies to settings where bids need to be placed before outcome of other auctions is known
\item applies to many settings, not just auctions, e.g. job applications, procurement contracts
\item simultaneous auctions effective way of quickly selling multiple items, e.g. computational resources
\item similar mechanism often used for e.g. spectrum licence auctions, although often this consists of multiple rounds. However, having multiple rounds is more susceptible to tacit collusion (famous account where bids were used as signalling devices).
\item lots of previous work on simultaneous sealed-bid auctions, both from OR and MAS literature, but equilibrium has largely remainded open problem, therefore proper evaluation of mechanism has not been possible:
\begin{itemize}
\item work by Rosenthal, e.g. \cite{szentes2003three} et al assumes essentially that players have the same preferences for the items (i.e. there are no unknown types)
\item work by Fatima et al, e.g. \cite{fatima2008bidding} assumes that bidders need to choose one of the auctions. However, Gerding et al have shown in \cite{ecs16075} that this is not optimal even for perfect substitutes (i.e. if everyone chooses only 1 of the auctions, then it is a best response under fairly general conditions to bid sctrictly positive in all auctions)
\item related is the work on competing auctions, but there the emphasis is on the seller or auction mechanism, and although in doing so they consider the bidder strategies as well, they similarly assume that bidders will only go to one of the auctions.
\item a number of papers, such as \cite{ecs16075} and \cite{ecs12600} allow bidders to bid in multiple auctions but only consider decision theoretic settings, not equilibria 
\item equilibria are analysed in \cite{krishna1996simultaneous}, but that's only for the case of complementary valuations and continuous bids. Our preference structure is more general and we consider discrete bids.
\item closest is our own previous work in \cite{ecs17271}, which also used FP to analyse simultaneous auctions, but only considers perfect substitutes. We significantly extend this work by adding theoretical analysis, consider more general preferences, provide extensive experiments and analysis of the strategies, and compare first and second price auctions. 
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